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Teaching video
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Graduate chapters and video availability01 · Read and understand
What you will learn
- Derive the pressure Poisson equation by taking divergence.
- Justify the conclusion "For u=(x,−y,0), f=0, Δp=−2; p=−(x²+y²)/2 works locally" using the stated assumptions.
Before you start
Incompressible momentum balance and second derivatives.
Keep paper nearby. Read the question once for the context, then again to identify what is known and what you need to find.
Start with a question
For smooth constant-density flow with normalized density one, derive the equation for p.
Why this math matters
Derive the pressure Poisson equation by taking divergence. This worked micro-lesson connects a precise mathematical condition to a conclusion you can check. The transfer task asks you to change the setting and decide which parts of the reasoning still apply.
Set up the model
A useful answer starts with clear assumptions:
- Fields are smooth and density has been normalized to one.
- Boundary conditions or a pressure gauge must be supplied in a global solve.
02 · Work through the example
Follow the reasoning, one step at a time.
Try to predict the next step before reading it. After each calculation, explain why the operation makes sense and how it helps answer the original question.
Work through it with Amar
See this example unfold.
The complete worked example, one idea at a time.
Recover pressure as a constraint-enforcing field
PausedQuestion: Start with the question. Paused.
Question
Start with the question
For smooth constant-density flow with normalized density one, derive the equation for p.
Before you calculate
Read what is known and what you need to find. Make a prediction before moving to the first calculation.
Starts paused. Play advances through the full text at a reading pace; pause whenever you need more time. Previous, Next, and the phase buttons let you set your own pace. Playback pauses when this walkthrough leaves the screen or you switch tabs.
Your device’s reduced-motion setting keeps each phase still. Manual controls remain available. The full written solution stays below.
Build the model
Take divergence of uₜ+(u·∇)u=−∇p+νΔu+f
Incompressibility kills divergence of the time derivative and the viscous Laplacian.
Work through the mathematics
Δp=−Σᵢⱼ(∂ᵢuⱼ)(∂ⱼuᵢ)+∇·f
Expanding divergence of convection and using ∇·u=0 yields the quadratic gradient source.
Check the conclusion
For u=(x,−y,0), f=0, Δp=−2; p=−(x²+y²)/2 works locally
Its pressure gradient balances the convective acceleration (x,y,0).
The result
For u=(x,−y,0), f=0, Δp=−2; p=−(x²+y²)/2 works locally
Its pressure gradient balances the convective acceleration (x,y,0).
Common mistakes to catch
- Pressure is not a freely chosen independent forcing after incompressibility is imposed.
- A local algebraic solution must not be mistaken for a global finite-energy example.
03 · Practice independently
Try it before revealing the answer.
Use paper or a calculator as needed. Write your units and reasoning, then open the hint or explanation to check your approach.
Practice 1
Does adding a spatial constant to p alter the velocity equation?
Show a hint
Differentiate the constant.
Reveal answer and explanation
No
Only pressure gradients enter momentum, so pressure needs a gauge choice.
Practice 2
Does this local polynomial example have finite energy on all R³?
Show a hint
Its velocity grows with position.
Reveal answer and explanation
No
It is a local identity check, not admissible finite-energy data on the full space.
Take the idea with you
Explain why solving incompressible flow couples distant regions through an elliptic pressure equation.
04 · Reflect and continue
Can you explain it in your own words?
Before moving on, explain the main idea without looking at the worked example. Try both practice questions, check your reasoning, and name one mistake you now know how to avoid. Return to a step if you still need support.
Up next: Compare forced and unforced equations without changing the theorem
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